Shape tilting in viscoelastic fluids

Transcription

1 Shape tilting in viscoelastic fluids Shape tilting refers to the fact that falling long bodies tend to line up along the longest line in the body. A cube ill fall in such a ay that the line beteen vertices is along gravity. F F Figure 1. Particles tend to line up vertex to vertex. The experiments of Lui and Joseph 1993 and Joseph and Liu 1993 give evidence to shape tilting, but not in such an ideal form as in the diagram. Joseph and Lui 1993 studied the sedimentation of cylinders in the viscoelastic fluid SI, hich is only modestly shear thinning. They say that, None of the cylinders turned broadside-on and all the round nose and cone end cylinders ould turn to put their long side straightly parallel to gravity. The sharp end cylinders tilt as they fall ith short cylinders tilting more than long ones (see figure 10). Liu and Joseph 1993 called this shape tilting, and they discussed some mechanisms associated ith extensional stresses at sharp corners hich could induce this tilting. [15] 1 DDJ/2001/proposals/microstructure/Tilt-Experiments.doc

2 (1) (2) (3) (4) (7) (8) (9) (5) Figure 2. Sketch of the orientation of cylinders falling in S1 in the channel of inches except particle (6) in the channel of inches. The particles (1) to (6) are tungsten carbide and (7) to (9) are brass. (6) Effects of shear thinning The tilting of falling long bodies is not restricted to shape tilting. The tilting of symmetric bodies falling in viscoelastic fluid is not yet ell understood. It is possibly associated ith all effects and the effects of shear thinning. Many experimenters look to study the effects of normal stresses ith shear thinning. The preferred experimental method for eliminating the effects of shear thinning is to use a fairly dilute solution of a high molecular eight polymers into a highly viscous Netonian solvent. These solutions are called Boger fluids. They have nearly constant viscosities and to exhibit normal stress effects. Hoever, the ratio of the normal stresses to shear stresses is rather small so that the fluids that are viscous dominate and falling particles tend to behave as they do in a Netonian fluid. It ould be desirable to understand the effects of shear thinning from theory so that experiments ith natural viscoelastic fluids, hich nearly alays shear thin, could be interpreted. Joseph gave a heuristic argument hich indicates that one important effect of shear thinning is to augment the effects of the normal stresses (the viscoelastic pressure) hich tend to turn the body into the stream. The heuristic argument shoing ho shear thinning amplifies normal stresses is as follos: consider the normal stress at a plane all and then imagine that a similar situation applies at the boundary of a long falling body. If the fluid shear thins then the all shear stress is given by 2 DDJ/2001/proposals/microstructure/Tilt-Experiments.doc

3 = ηγ γ and the viscosity γ τ thins). But the normal stress η goes don hen the shear rate γ goes up (shear 2 τ ψ 1 γ = γ (2) ηγ must strongly increase if τ is constant as it ould be, say in the Poiseuille flo in a pipe. Huang, Hu and Joseph 1998 shoed that ellipses falling in a viscoelastic fluid ere stable in a tilted orientation hen the fluid shear thins and ere unstable hen shear thinning as suppressed. It ould be of great interest to have mathematically rigorous results for the effects of shear thinning. Experiments Particle pre-merger Mirror (45 degree) Y X Particle Z Particle recovery screen Figure 3. The sketch of the sedimentation channel. Channels In the sedimentation experiments particles ill be dropped in a liquid-filled channel. To channels ill be made, one called a to-dimension channel (0.5 x 8 x 40 ) and another called a three-dimensional channel (4 x 4 x 36 ). The to-dimensional channel is basically threedimensional though one dimension of the cross section is much smaller than the other. In the 3 DDJ/2001/proposals/microstructure/Tilt-Experiments.doc

4 to-dimensional channel, there is a thin and long mirror places in 45 degree in one side, so that both the front and side tilting can be observed in front vie. Also paper scales graduated to one millimeter are fixed to the backside of the channel. In the to-dimensional channel, the distance beteen the dropping particles and the front and back all is very small. The effects of the sidealls ill be very important. The three-dimensional channel is set up to minimize the side all effects. Particles are collected on a screen that may be ithdran ithout changing the fluid. In order to avoid air bubbles on the particles, a pre-etting device ill be used. Particles Various particles, such as cylindrical particles, cubic particles and plate particles ith different eights ill be used in the sedimenting test. In order to get systematic results for samediameter cylindrical particles, flat ends, round ends and flat angle cut ends ith the same eight ill be made. We ill make the same particle of different material to vary the eight for a fixed shape. Measurement and visualization Velocities and tilt angles ill be measured ith a high-speed digital camera hich can take pictures at 1000 frames per second. The images can be replayed forard and backard at different play rates, and also can be stored in computers. Movable reticles allo spatial measurement, and the elapsed time is observed hile the recording is being made and replayed. Those functions allo one to measure the falling speed and tilt angle of sedimenting particles. To visualize particle paths, orientation and fluid streamlines, color dye ill be injected to the channel to form horizontal strip lines. These lines form a grid hich deforms under motion and they give rise to an excellent record of the streamlines and ake structures created by falling particles. 4 DDJ/2001/proposals/microstructure/Tilt-Experiments.doc

5 References 1. A. Fortes, D.D. Joseph, and T.S. Lundgren. Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech. 177, (1987). 2. R. Gloinski, T.-W. Pan, T.I. Hesla, D.D. Joseph, and J. Périaux. A fictitious domain method ith distributed Lagrange multipliers for numerical simulation of particulate flos, in Domain Decomposition Methods, (J. Mandel, C. Farhat, X.C. Cai eds.), American Mathematical Society, Providence, R.I., 10, , (1998). 3. H.H. Hu. Simulation of particulate flos of Netonian and viscoelastic fluids, accepted for publication in Int. J. Multiphase Flo (1998). 4. H.H. Hu. D.D. Joseph, and A. Fortes. Experiments and direct simulations of fluid particle motion, Int. Vid. J. Eng. Res. 2, 17 (1992) 5. P.Y. Huang, J. Feng, H. H. Hu, and D.D. Joseph. Direct simulation of the motion of solid particles in Couette and Poiseuille flos of Viscoelastic Fluids, J. Fluid Mech. 343, (1997). 6. P.Y. Huang, J. Feng, and D.D. Joseph. The turning couples on an elliptic particle settling in a vertical channel, J. Fluid Mech. 271, 1 16 (1994). 7. P.Y. Huang, H. H. Hu, and D.D. Joseph. Direct simulation of the sedimentation of elliptic particles in Oldroyd B fluids, J. Fluid Mech. 362, (1998). 8. D.D. Joseph. Finite size effects in fluidized suspension experiments, in Particulate To- Phase Flo, (M. C. Roco, ed.), pp Butterorth-Heinemann (1993). 9. D.D. Joseph. Flo induced microstructure in Netonian and viscoelastic fluids. In Proc. 5 th World Congress of Chem. Engng, Particle Technology Track, San Diego, July AIChE, 6, 3-16 (1996). 10. D.D. Joseph and J. Feng. A note on the forces that move particles in a second order fluid, J. Non-Net. Fluid Mech. 64, (1996). 11. D.D. Joseph, A. Fortes, T.S. Lundgren, and P. Singh. Nonlinear mechanics of fluidization of beds of spheres, cylinders and disks in ater, in Advances in Multiphase Flo and Related Problems, (G. Papanicolau, ed.), pp SIAM (1987). 12. D.D. Joseph and T. Liao. Potential flos of viscous and viscoelastic liquids, J. Fluid Mech., 265, 1-23, (1994). 13. Joseph, D.D. and Liu, Y.J Orientation of long bodies falling in a viscoelastic liquid. J. Rheol. 37, D.D. Joseph, Y.J. Liu, M. Poletto, and J. Feng. Aggregation and dispersion of spheres falling in viscoelastic liquids, J. Non-Net. Fluid Mech. 54, (1994). 15. L.G. Leal. Particle motions in a viscous fluid, Ann. Rev. Fluid Mech. 12, 435 (1980). 16. Liu, Y.J. and Joseph, D.D Sedimentation of particle in polymer solutions. J. Fluid Mech. 255, M. J. Riddle, C. Narvaez, and R. B. Bird. Interactions beteen to spheres falling along their line of centers in a viscoelastic liquid, J. Non-Net. Fluid Mech. 2, (1977). 18. P. Singh, P. H. Caussignac, A. Fortes, D.D. Joseph, and T. S. Lundgren. Stability of periodic arrays of cylinders across the stream by direct simulation, J. Fluid Mech. 205, (1989). 5 DDJ/2001/proposals/microstructure/Tilt-Experiments.doc

6 Numerical simulation The goal of our numerical simulations is to simulate experiments and suggest hypotheses about migration and lift of spherical particles in shear flos and the orientation of long particles and shape tilting in sedimentation. For this purpose e plan to sue the distributed LaGrange- Multiplier (DLM) method. The basic idea of this method is to imagine that fluid fills the space inside as ell as outside the particle boundaries. The fluid-flo problem is then posed on a larger domain (the fictitious domain ). This larger domain is simpler, alloing a simple regular mesh to be used. This in turn allos specialized fast solutions techniques. The larger domain is also time-dependent, so the same mesh can be used for the entire simulation, eliminating the need for repeated remeshing and projection this is a great advantage, since for three-dimensional particulate flo the automatic generation of unstructured body-fitted meshes in the region outside a large number of closely spaced particles is a difficult problem. In addition, the entire computation is performed matrix-free, resulting in significant savings. The velocity on each particle boundary must be constrained to match the right-body motion of the particle. In fact, in order to obtain a combined eak formulation ith the hydrodynamic forces and torques eliminated, the velocity inside the particle boundary must also be a rigid-body motion. This constraint is enforced using a distributed LaGrange multiplier, hich represents the additional body force per unit volume needed to maintain the rigid-body motion inside the particle boundary, much like the pressure in incompressible fluid flo hose gradient is the force required to maintain the constraint of incompressibility. DLM has been implemented [refxx18 ] for viscoelastic fluids. A solution for the collisions has only recently been found (Singh et al 2001) DLM orks ell for three-dimensional simulation (Pan, et al 2001) but is too expensive and has not been fully parallelized. All routines in our three-dimensional DLM codes are parallel except those that deal ith solutions of the Poisson problem. On an eight-processor machine e are currently getting a speedup of around 4.5 and the bottleneck comes from the parts that are currently not parallelized. During the next fe months, e plan to rite parallelized Poisson solvers for our to and three DLM codes hich is expected to increase the speedup factor to 6.5 or better. We also plan to test the fully parallelized code on more than eight processor machines to determine if the speedup remains satisfactory hen the number of processors used ins much larger than eight. If e find that the speedup factor deteriorates significantly, e ould use other approaches in addition to the OpenMP hich e are currently using to obtain the desired speedup factor. 1 DDJ/2001/proposals/microstruct/NumSim-Expmts.doc

7 References 1. E.S. Asmolov. The inertial lift on a spherical particle in a plane Poiseuille flo at large channel Reynolds number, J. Fluid Mech. 381, (1999). 2. Bagnold, R. A., Fluid forces on a body in shear-flo; experimental use of stationary flo, Proc. R. Soc. Lond. A., 20, (1974). 3. H. Brenner. Hydrodynamic resistance of particles at small Reynolds numbers. Adv. Chem. Engng, 6, 287, (1966). 4. P. Cherukat and J. McLaughlin. The inertial lift on a rigid sphere in a linear shear flo field near a flat all, J. Fluid Mech. 263, 1 18 (1994). 5. Cherukat, P., J. B. McLaughlin, and A. L. Graham, The inertial lift on a rigid sphere translating in a linear shear flo, Int. J. Multiphase Flo, 20, (1994). 6. R.G. Cox and S.G. Mason. Suspended particles in fluid flo through tubes, Ann. Rev. Fluid Mech. 3, 291, (1971). 7. Eichhorn, R. and S. Small, Experiments on the lift and drag of spheres suspended in a Poiseuille flo, J. Fluid Mech. 20, (1964). 8. F. Feuillebois. Some theoretical results for the motion of solid shperical particles in a viscous fluid, in Multiphase Science and Technology (ed. G.F. Heitt et al.), 4, 583 Hemisphere, (1989). 9. Graham, A. L. and R. B. Bird, Particle clusters in concentrated suspensions. 1. Experimental observations of particle clusters, Ind. Eng. Chem. Fundam. 23, (1984). 10. H.H. Hu, D.D. Joseph, Lift on a sphere near a plane all in a second-order fluid, J. Non-Netonian Fluid Mech., 88, King, M. R. and D. T. Leighton, Jr., Measurement of the inertial lift on a moving sphere in contact ith a plane all in shear flo, Phys. Fluids, 9, (1997). 12. Krishnan, G. P. and D. T. Leighton, Jr., Inertial lift on a moving sphere in contact ith a plane all in a shear flo, Phys. Fluids, 7, (1995). 13. L.G. Leal. Particle motions in a viscous fluid, Ann. Rev. Fluid Mech. 12, 435 (1980). 14. McLaughlin, J. B., Inertial migration of a small sphere in linear shear flos, J. Fluid Mech. 224, (1991). 15. P. Singh, D.D. Joseph, T.I. Hesla, R. Gloinski, T.W. Pan, A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flos, J. Non-Netonian Fluid Mech., 91, T.W. Pan, D.D. Joseph, R. Bai, R. Gloinski, V. Sarin, Fluidization of 1204 spheres: simulation and experiment, J. Fluid Mech., Accepted. 17. P. Singh, T.I. Hesla, D.D. Joseph, A modified distributed Lagrange multiplier/fictitious domain method for particulate flos ith collisions, Int. J. Multiphase Flo, submitted. 2 DDJ/2001/proposals/microstruct/NumSim-Expmts.doc